What Is It to Be Real? Numbers as Real Species of a Category in the Late Medieval Debate about the Ontological Status of Numbers

3.1 Introduction

Among the conceptualist anti-realists, the most common reaction to the realist argument sketched out above, that number belongs to a (real) category, and only what is a real being can belong to a (real) category, is to simply deny its major premise. As the Franciscan Peter Auriol (ca. 1280–1322) puts it succinctly:

A category is not called real because everything that is contained within it has being outside the soul.^[25]

3.2 John Duns Scotus

As Maria Sorokina has observed, the foundations for this reply were laid by the Franciscan John Duns Scotus (1265/6–1308), who states that, both according to Aristotle and according to the truth, one and the same category can contain both real beings and beings of reason. Thus, in addressing the issue of how it is possible that a number is not a real being if it is a member of the category of Quantity, Scotus explains that Quantity comprises both quantity with extramental existence, the Continuous Quantity, and quantity which, at least as far as its form is concerned, has mental being only: the Discrete Quantity. The soul, by counting, measures a given discrete quantity and gives it its name; this name signifies primarily the measure in the soul, which is the form of number, and which can then be denominatively predicated of the multitude of counted things existent in reality.^[26]

3.3 Henry of Harclay

Henry of Harclay (1270–1317), who was Scotus’s student and follower, expands on the ideas originally found in Scotus. He rejects the major premise of the realist argument, namely, that the form of an item belonging to a real category must be itself real. He gives counterinstances to this premise: for example, not only real relations but also relations of reason (ones where one or both terms do not have real existence) belong to the real category of Relation.^[27] Similarly, not all species of Quality are real: for instance, shape is not, and is in fact reducible to continuous quantity. Shape is classified as a quality only secundum dici (presumably because it responds to the question of what a thing is like, which is standardly a question seeking quality – this is a reply given by many later Scotists^[28]), but in re it is identical with continuous quantity because it is nothing more than the “closure of lines containing continuous quantity.”^[29]

Henry holds that just as both mere beings of reason (relations of reason) and real beings (real relations) belong to the category of Relation, so the category of Quantity consists of the genus of Continuous Quantity, which has under itself real things, that is, things existing in the extramental reality, and the genus of Discrete Quantity, which has under itself, among others, time and number, which have no extramental existence but only mental existence, since they exist objectively in the mind.^[30]

3.4 Francis of Marchia

The Franciscan Francis of Marchia (ca. 1285/90–after 1344) expands the point (taken either from Henry of Harclay or, what is more likely, from Auriol, who himself borrowed it from Henry) about relations. The challenge that Auriol, Francis of Marchia, and others, raise against the realists is this. All the thinkers that I am familiar with do recognize that, despite the difference in their ontological status, both types of relation fall under the same category, namely, Relation itself. Thus, Francis points out that if the realists wanted to stick with their original major premise, that only what is a real being falls under a category, they would need to deny the existence of relations of reason or insist that all relations are real relations:

<To the argument> that number is a species of Quantity, which is a real genus, I reply that if this were a good argument, it would follow that no relation in the world would be a relation of reason, because every relation of reason, or at the very least some, is a species of Relation, which is a real genus.^[31]

Thus, Francis emphasizes, real relation and relation of reason are two species belonging to the same genus, that is, to the same category of Relation. This shows that a certain class of entities can belong to a category without possessing extramental existence. Thus, going back to Quantity: even if one concedes that the other basic species of Quantity, Continuous Quantity, is real, this will have no import for the ontological status of Discrete Quantity, just like in the above case the status of real relations has no import for the status of relations of reason. Thus, the anti-realists add, on this understanding, analogous to the case of real relations and relations of reason, Continuous Quantity, a genus of real being, and Discrete Quantity, a genus of being of reason, will be two genera which together constitute one category, despite the fact that number has no real, extramental being.^[32] This presents the realists with a serious dilemma, to which, I think, they have no satisfying reply: if they admit that relations of reason, despite their lack of extramental existence, are species of the genus of Relation, why does number, based on the fact that it is a species of the genus of Quantity, have to be a real being?

3.5 Evaluation

One of the most basic problems with these objections is, of course, their heavy reliance on analogies with other cases which the anti-realist authors take to involve categories containing both real beings and beings of reason. For the realist authors could simply deny that, for example, a shape is not a real being. Indeed, there is even disagreement about this issue among the authors using the argument: as I have said above, Henry of Harclay lists shape as an example of an item in the category of Quality that is not a real being. By contrast, Peter Auriol states explicitly that shape is a real being.^[33]

But, setting these quibbles about shape and so on aside, there remains a much more serious problem for the realists, which was pointed out by their opponents: that of real relations and relations of reason. Most late medieval thinkers, regardless of their ontological sympathies, do recognize the distinction between real relations and relations of reason.^[34] On the standard medieval way of conceiving of a relation, each relation has two terms: its foundation and its terminus. For example, if Socrates is similar to Plato because both Socrates and Plato are wise, then Socrates’s wisdom is the foundation, and Plato’s wisdom is the terminus, of the relation of similarity existing between Socrates and Plato.^[35] Moreover, on the standard account, a relation is real if both its foundation and its terminus are real. By contrast, a relation of reason does not require the reality of both of its terms. If, for example, I am thinking of a golden mountain, there is a relation of thinking of existing in me and the golden mountain, but only one of its terms, namely, me, is real, whereas the golden mountain is a mere being of reason. Thus, the case of real relations and relations of reason seems to undermine the premise that the categories encompass real being only, and thus to undercut the whole realist rationale for realism about numbers based on categorial belonging.

3.6 Aftermath

As a result of the above considerations, some thinkers, such as Auriol and Francis of Marchia, are led to the rejection of the major premise of the original realist argument that they criticized, namely, that only real being can belong to a real category, and thus allow for all kinds of categorial items that are mere beings of reason. They are happy to concede that, notwithstanding that number is merely a being of reason, it is nevertheless a species of the category of Quantity, just as a relation of reason is a species of the category of Relation.

But other thinkers take a different, somewhat more radical route, because they deny that number is properly speaking a species of Quantity at all. For example, the Franciscan Scotist Aufredus Gonteri Brito (fl. ca. 1320) explicitly states that in the Categories, Aristotle enumerates kinds of Quantity, including number, “which are not per se quantities,” that is, which do not belong per se to the category of Quantity and thus cannot constitute one of its species. Many medieval Scotists are rather skeptical about the Aristotelian classification of the ten categories. For example, many of them hold, against Aristotle, that properly speaking, shape, which Aristotle classifies in the Categories as the fourth species of Quality, is in fact a relational accident.^[36]

I have made this digression for a purpose, since Aufredus uses his description of the twofold classification of shape to illustrate the status of number. He says that “other <items>, such as number, time, and place, are called quantities <merely> in respect of their concept and manner of speaking (secundum rationem et dici); nor did Aristotle mean otherwise.”^[37] This of course still leaves the question open as to what, if anything, numbers are in re, and unfortunately Aufredus never fills this gap in his account. One can only suspect that he would perhaps go on to say that in reality they are reducible to the numbered things.

Why, though, given that, on Aufredus’s account, number is not a quantity in re, did Aristotle, rather misleadingly, describe it as if it were? Here Aufredus presents us with two explanations, one more specific and one more general. Aufredus’s specific explanation relies heavily on his detailed account of what number is, which I can only here sketch very briefly. Following Henry of Harclay, Aufredus thinks that a number consists in a replication of unities performed by the intellect and in the intellect: I start with one and replicate it as many times as needed in order to obtain a given number. Now, Aristotle says in Book X of the Metaphysics that a characteristic feature of quantity is that it is a measure.^[38] Weaving these threads together, Aufredus tries to show how his concept of number as a replication of unities and the concept of number as a measure could (mis)lead some into believing that number belongs to the category of Quantity:

If we want to explain […] why number is posited by Aristotle to be a quantity, the reason is as follows: Because the nature of measure belongs first of all to quantity, as is said in Book X of the Metaphysics, for this reason, because unity considered by the intellect or imagination by its replication can measure the continuum that is divisible in diverse parts, for this reason such a replication and comparison of distinct part or parts performed by the comparing intellect is called Quantity, and such a replication of unities is nothing else than a number, which is a measured multitude, as is said in Book X of the Metaphysics, that is, in reality a number is a multitude or continuity divided into parts measured by one and by the replication of one.^[39]

There is an easy way to translate this rather convoluted passage into simpler English, which is to go back to the case of shape. As I have mentioned, one of the usual reasons given by thinkers who reclassified shape from Quality (where it had originally been put by Aristotle) to one of the relational categories (usually Position) was that shape is quality de dici because it responds to the question of what something is like, which is the question normally seeking quality. In a somewhat analogous way, Aufredus explains that because the proper function, as it were, that Aristotle attributes to Quantity is being a measure, and a number seems to be a measure (and thus, albeit this remains implicit in the passage, a number also provides a reply to the question of ‘how many,’ which seeks Quantity), it looks as if number is, properly speaking, a species of the category of Quantity.

This was the more specific reason why, according to Aufredus, Aristotle described number in a way that might suggest that it is a quantity. What is the more general reason? Here Aufredus borrows his idea from Simplicius, who, in struggling with explaining and reconciling various theses expressed in Aristotle’s Categories, said about the theses that he found impossible to explain that, because the Categories are an introductory work to Aristotle’s logic, and thus the whole of his philosophy, Aristotle sometimes includes in them the views commonly held by others that he himself does not share but does not (yet) attempt to disprove. Aufredus claims that this may be the case with numbers, namely, that Aristotle borrowed the classification of number as quantity from others, as a popular view, without himself espousing it. (Needless to say, there is absolutely no textual evidence in Aristotle for this being so.)^[40]

3.7 Simpler Solution? Francis Marbres

A solution that is at least partially different, and perhaps also somewhat simpler compared to the ones sketched out above, was proposed by Francis Marbres (d. ca. 1330). Francis’s reply focuses on what he takes to be different senses of being real: a stronger sense, which he thinks does not apply to numbers, and a weaker sense, in which he thinks numbers can be called real, but which does not require one to posit that their form has extramental existence. After laying out and endorsing a conceptualist view of numbers, Francis is confronted with the by now familiar objection: that the species of a real category must have real being, that is, being in extramental reality, and since number is a species of the category of quantity, it must have extramental existence and hence cannot be made by the mind.^[41] In response to this objection, Francis introduces a distinction:

I respond by saying that that something is real [aliquid esse reale] can be understood in two ways.

1. In one way, <something can be real> because it has being regardless of any act of the intellect and because it does not depend upon the soul either objectively or effectively; for example, a stone and others of this kind <have this kind of reality>.^[42]

This is the strong sense of being real, or having real being, which is completely independent of the existence or act of the intellect. A thing possessing this kind of real being is independent of the intellect both objectively – that is, it does not need to be the object of thinking of the intellect – and effectively – that is, it does not need to be made up by the intellect. A stone is a good example, since, while I might think of it (in this way it will have objective existence in my mind), it will still be out there regardless of whether I or anyone else think of it at any time. This, as one can expect, is not the way in which the reality of number should be understood according to Marbres. For, he says, there is another way of conceiving of it:

1. In another way, <something can be real> because it depends on the soul objectively but not effectively; for example, a number.

2. Or, if it depends on the soul effectively, it nevertheless informs the intellect and is its proper accident; for example, an act of understanding and of knowing.^[43]

Marbres makes it clear that for a species of a real category to be real, or for that species to even qualify as a suitable candidate for a species of a real category, it is not necessary that it be real in the first sense but it suffices that it be real in the second sense.^[44]

In the above passage, Marbres introduces a further sub-distinction within categorial items with mental existence: (2a) some of them depend on the soul objectively only but not effectively (see above for what this jargon means), whereas (2b) others depend on the soul effectively. Both the structure of the passage, which is clearly split into two distinct parts, and the fact that each part concludes with a different example illustrating a given type of being real, where numbers only serve as an example for (2a), makes clear that according to Marbres numbers fall under (2a) only.

My interpretation is also supported by how Marbres responds to a different realist argument. In a nutshell, the realists argue that a real science must be concerned with what is real; and given that arithmetic is a real science (rather than science occupied merely with our thinking), its object must be real; so numbers must have their own reality. In his reply to this argument, Marbres not only emphasizes that what suffices for a given item to be an object of a real science is that it have objective being in the soul, but he is equally emphatic that this is the only kind of being in the soul which is sufficient for number to be real and so be an object of a real science: if number had soul as its efficient cause, it could not be an object of a real science, presumably because then it would be made up by the mind. Number can be a species of a real category because it “depends on the soul objectively but not effectively.”^[45] This means that the soul does not make up numbers but only contains forms of numbers.

This is interesting because it shows that Marbres is insistent on preserving some degree of mind-independence of numbers: he emphasizes that if the mind were their efficient cause, they would not be real and so could not be a species of a real category. Worth adding is that the objective being that numbers have in the mind, while it of course does involve the mind, is nevertheless tied to reality, to grasping the multitude of objects presenting themselves via our senses to our mind.

3.8 A Nominalist Take: William of Ockham

The debate summarized above was based on various versions of the broader assumption that the categories are a classification of all that exists; in other words, that they carve up exhaustively the whole of reality. Of course, this means that thinkers who do not share this assumption consider the above to be a pseudo-problem: if the categories are an exhaustive list but merely of the ways we speak or think about reality, then, while of course reality taken as a certain whole remains the basis for the categorial scheme (after all, as I have said, these would be ways of speaking or thinking about reality), still the categorial items would not need to satisfy any criterion of reality in order to be counted as such.

One of the most famous representatives of this position is, of course, the Franciscan William of Ockham (ca. 1287–1347). When, in his discussion of number in his Sentences commentary, Ockham attacks the alternative, realist views of numbers, and, what is now of most relevance, the realist argument based on categorial identity of number, he begins by giving a brief summary of the relevant aspects of his view of the categories:

Some […] would say^[46] that some of the categories do not signify things distinct from other things belonging to other categories, but that instead <these categories> signify the same things by different modes of signification. For example, according to some, the category of relation does not signify a thing distinct from the absolute thing of another category but rather signifies that <absolute> thing of another category together with connoting one other thing, either of the same category or of a different category.^[47]

This is admittedly a rather complicated passage, but in essence what Ockham means to say is that, with the exception of some categories (according to him, Substance, some of the species of Quality, and relations in the Trinity),^[48] (a) the other categorial names do not stand for a thing that would be different from other things in the world or from the other categories and that (b) the categories are distinct from each other by having different modes of signification. In the case of relations, mentioned in the passage above, in order to provide an exhaustive account of the ontology of relations, say, of how it is that X is similar to Y, one does not need to posit some third thing over and above X and Y. If a realist about, say, the relation of similitude wanted to force me into admitting that if Plato is similar to Socrates, this can only happen in virtue of similitude, which is an extra thing over and above Plato, Socrates, and, say, the qualities that are compared (say, wisdom), I could reply that the statement that Plato is similar to Socrates signifies Plato (and the relevant quality being the source of similitude, such as wisdom) while connoting, that is, signifying secondarily, Socrates (and his quality of wisdom); this is all that is needed for Plato to be similar to Socrates, without positing any intermediary thing, as it were, that would somehow point from Plato to Socrates. Thus, the peculiar mode of signifying of similitude is signifying one quality while connoting another of the same kind; say signifying Plato’s quality of wisdom while connoting Socrates’s wisdom.^[49]

So far, so good, but how does this apply to number, which is a species of Discrete Quantity? According to Ockham, the explanation of the characteristic mode of signification of number, and of Discrete Quantity in general, on the account that he endorses is that “discrete quantity would be a concept signifying things themselves and connoting that they do not make up something one per se.” In other words, a numerical term signifies the numbered things while connoting their discretion, that is, the fact that they are not united into a single per se whole. The species of number, then, can never signify an individuum but rather always signifies a plurality of things or of terms that stand for a plurality of things.^[50]

This is not a question that Ockham ever dealt explicitly in his texts, but one can easily see that a realist opponent could raise the following objection to his account: If a number is just a concept or a term that we apply to a multitude of things while thinking of them as discrete from one another, what basis do we have for assigning a given concept or term to a given multitude? For example, if I apply the concept or term ‘four’ to the chairs standing in my room, what is the truthmaker of the mental or verbal proposition ‘There are four chairs in my room’? Either this imposition is completely arbitrary, and I could assign any other numerical concept or term just as I please, or there must be something in reality that would make my proposition true (or false). Ockham, I am sure, would agree with the latter statement, just as he does in so many other contexts.^[51] But he would refuse to make the key step that the realists deem inevitable, namely, to say that there is a further thing in reality that makes the chairs in my room be four in number and thus serves as the truthmaker of my proposition about their number. Rather, Ockham would insist that there being a specific multitude of chairs in my room is a brute fact which cannot be explained any further.

To summarize, Ockham is convinced that in order to explain the categorial identity of numbers, it suffices to employ the notion of the mode of signification that is peculiar to each category, in this case to Discrete Quantity; and nothing more is needed in order to make number belong to such a species, for reasons (at least partly) laid out above. Of course, as I have said, this is not so much a solution of the original problem as it is a dismissal of it, which stems from a radically different approach to the status and distinction of the categories. But this broader topic lies far beyond the scope of the present study, so I shall leave it at this point.^[52] Ultimately, it seems to me, because the original source of disagreement between Ockham and the realists about numbers lies so deep, namely, it concerns the status of the Aristotelian categories, it is difficult to assess how well Ockham’s view of number fares compared to the realist account. Rather, I suggest, it is best to think of these two as two very different, and incompatible, paradigms.